Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #45020 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,14)(11,19,20,16)(12,22,21,17)(13,18,23,24)(25,26,30,29)(27,31,35,32)(28,36,38,33)(34,40,37,39);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,14,19)(12,13,21,23)(17,18,22,24)(25,27)(26,31)(28,34)(29,32)(30,35)(33,39)(36,40)(37,38);(11,20)(13,23)(16,19)(18,24)(25,27,30,35)(26,32,29,31)(28,37,38,34)(33,39,36,40);(1,2,5,4)(3,6,8,7)(9,12,15,21)(10,17,14,22)(11,13,20,23)(16,18,19,24)(25,27)(26,32)(28,34)(29,31)(30,35)(33,40)(36,39)(37,38);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,21)(12,20)(14,24)(15,23)(16,22)(17,19)(25,28)(26,33)(27,34)(29,36)(30,38)(31,39)(32,40)(35,37);(9,14,15,10)(11,16,20,19)(12,22,21,17)(13,18,23,24)(25,29,30,26)(27,31,35,32)(28,33,38,36)(34,40,37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895334);